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%I #10 Jan 08 2022 08:57:06
%S 1,0,1,1,1,0,1,1,3,2,4,4,5,3,3,4,6,2,7,6,13,9,19,16,27,21,40,33,47,37,
%T 54,48,66,51,65,65,77,64,80,71,96,60,106,95,112,93,152,114,191,131,
%U 242,192,303,210,366,300,482,352,581,450,713,539,882,689,995
%N Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.
%C An integer partition is knapsack if every distinct submultiset has a different sum.
%C These are the subsets counted by A325867, ordered by sum rather than maximum.
%H Fausto A. C. Cariboni, <a href="/A326015/b326015.txt">Table of n, a(n) for n = 1..600</a>
%e The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):
%e {1} {2,1} {3,1} {3,2} {4,2,1} {5,2,1} {4,3,2} {6,3,1} {5,4,2}
%e {5,3,1} {7,2,1} {6,3,2}
%e {6,2,1} {6,4,1}
%e {7,3,1}
%e .
%e {5,4,3} {6,4,3} {6,5,3} {6,5,4} {7,5,4} {7,6,4}
%e {7,3,2} {6,5,2} {8,5,1} {7,6,2} {9,4,3} {9,5,3}
%e {7,4,1} {7,4,2} {9,3,2} {8,4,2,1} {9,6,1} {9,6,2}
%e {8,3,1} {7,5,1} {9,4,2,1} {8,4,3,2}
%e {9,3,1} {9,5,2,1}
%e {10,4,2,1}
%t ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]]
%t maxsks[n_]:=Select[Select[IntegerPartitions[n],UnsameQ@@#&&ksQ[#]&],Select[Table[Append[#,i],{i,Complement[Range[Max@@#],#]}],ksQ]=={}&];
%t Table[Length[maxsks[n]],{n,30}]
%Y Cf. A002033, A108917, A275972, A276024.
%Y Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.
%K nonn
%O 1,9
%A _Gus Wiseman_, Jun 03 2019
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